Originally, Lucan was hunted down because his former University felt him a danger to society and wanted him to be Institutionalized. The plot shifts in the latter part of the series when Lucan became a fugitive after a tragic incident at the University.
One night he catches two doctors breaking into a lab to steal drugs. A fight ensues and a chemical fire erupts. One of the doctors escapes with a badly burned arm. The other suspect died on the scene, but not before he blamed Lucan for the fire and theft. The perpetrator and flashbacks to that night are revealed in an excellent episode called “Nightmare.”
Lucan escapes the University before he goes to trial. A Bounty Hunter named Prentiss follows Lucan wherever he goes – just like his television predecessor Lieutenant Gerard from The Fugitive, and similar to the abrasive Journalist Mr. Jack McGee on The Incredible Hulk.
The fugitive storyline was incorporated toward the end of the series. Before that, Lucan was simply trying to find his parents and experience the outside world. The plot change is that Lucan escaped the University and they still own him by court order and claim he’s a danger to society. They sent Prentiss to bring him back for caged observation.
The writers clearly couldn’t decide which backstory to use and the show had three versions of opening credits. Rolled into one they are all plausible, but the “Fugitive” angle is an unfortunately over-used TV Trope. (I personally never get tired of it. My guess is the Showrunners or Producers felt that the endearing story of Lucan, a young man trying to find his place in the world, needed a jolt. So Prentiss is the predator and Lucan his prey.